The general reason for the persistent belief in the blood libel is that once the meme appearead, it became immortal and traveled from village to village and from country to country as a rumor in which at first the ignorant peasants, various travelers and monks and then the others would believe. In later times, after printed books appeared, whole tomes with the blood libel accusations, gathering the hitherto known cases, were massively produced, which further helped to spread the blood libel also among the more educated parts of the population.

So when anything happened to a Christian child, the Jews, "the Other", were the first to blame, since they had already been under suspicion.

Add to this that various kings and other nobility would use the threat of the blood libel to get rid of the Jewish communities, to whom they might have owed significant sums of money. Exterminating or expelling them would sometimes be easier than paying back the debts.

The common Christian belief in the "Christ-killing" Jew, the "cursed" people, also helped the meme to survive. It was argued that the "cursed" people needed the "blessed" Christian blood to survive at all, to heal their diseases that come from the "curse", etc. Though ironically it was the Christians themselves who were used to be accused by the Romans of ritual cannibalism, eating babies and whatnot.

As to why it was believed despite the religious prohibition: that the orthodox rabbinical Judaism has such a general prohibition doesn't mean that those who believed in the libel a) knew about it; b) believed that it applied in this particular case. There's also a prohibition on murder in Judaism, but that didn't exactly convince the mobs.

Now, in the case of the ignorant peasants etc. the reason is clear, they didn't exactly care about logic, were generally ignorant and prejudiced against the Jews, and could accuse the Jews as a whole of such blood rituals regardless of the actual practices.

However, and particularly in the Russian empire, it was often argued by the slightly more sophisticated antisemites, that not the whole of Judaism believed in the killing of Christians and consumption of their blood, but some sort of a specific pernicious Judaic sect that must have hidden itself among the Jews. Explaining that the existence of such a sect was extremely unlikely given the close-knit nature of the Jewish communities (and thus the accusation was effectively against all Jews) was, of course, for nought.

The last person who resurrected this argument was, paradoxically, the Israeli historian Ariel Toaff, who in his widely condemned book Passovers of Blood argued for some core of truth behind the libel, based on the tortured confessions (which alone made his thesis invalid, for, although this sort of confessions is sometimes used by the historians to reconstruct social history, it is useless when it comes to the alleged crimes themselves - the very point of the torture), as well as the fact that some Jews, along with their gentile neighbors, succumbed to the widespread medieval folk belief of the curative power of both animal and human blood (the latter of which could be acquired through medicinal bloodletting, voluntary donors etc.), in this case when consumed in dried out and dissolved form (in wine, water, potions etc.), something that Toaff richly documents by citing both the "low-level"/basicaly magical literature of the "practical Kabbalah" (about animal and human blood) and even some rabbinic responsa (strictly in relation to the dry animal blood). Indeed, we still find the traces of such beliefs in regard to the animal blood here and there:

https://www.ou.org/torah/halacha/hashoneh-halachos/fri_06_22_12/

If a person eats food dipped in a liquid, or if the liquid gets on the food and it is still wet, one must wash his hands without a blessing even if he doesn’t touch the wet part. [...] There are seven liquids in this matter: wine (including wine vinegar), bee honey, olive oil, milk (including whey), dew, blood of an animal or bird, and water. Blood may not normally be eaten but this applied when it might have been used as part of a medicinal cure.

https://www.chabad.org/library/article_cdo/aid/3323451/jewish/Shulchan-Aruch-Chapter-158-Rules-Governing-Washing-Ones-Hands-for-a-Meal.htm

According to the Torah’s terminology, the term “liquid” refers only to seven liquids: water, wine, oil, milk, dew, bee-honey, and the blood 44 of domesticated animals, undomesticated animals, and fowl, i.e., one dips food in their blood for medicinal purposes, because of a life-threatening illness. 45

44 Rambam, Hilchos Tuma’as Ochalin 10:3, states that this refers only to the blood that flows at the time an animal is slaughtered. Other blood does not have this distinction.

45 Otherwise, partaking of their blood would be forbidden by a Scriptural prohibition.

So the general prohibition on the consumption of blood had an exception, and this could be a reason why it was not always seen as an effective counterargument since another - secret - exception could be argued for.

Finally, many antisemites will naturally argue that Jews are liars, which is why proffering such arguments won't have much effect on them anyway. The Nazis like Julius Streicher railed against the kosher slaughter (intended to rid the meat from blood as fully as possible) and at the same time wailed about Christians babies "killed by Jews" to consume their blood, seeing no contradiction here.

Some of them would explicitly use the blood libel to further the murder of the Jews. We don't know, for example, whether Himmler personally believed in the blood libel, but we know that he instructed Kaltenbrunner to supply the Einsatzgruppen with the literature on the "Jewish ritual murder".

As to why some would still believe it - aside from basic antisemitism, look at what kinds of conspiracy theories are thrown around nowadays (Pizzagate, QAnon and such), involving secret cabals abusing and murdering children. Hundreds of thousands, maybe millions believe in these claims, which are not far away from the blood libel. Such beliefs don't always include antisemitic claims, but they're natural cousins to such claims, so you will often see the two combined ("elites are abusing children" + "elites are Jews"). And CTs like Pizzagate arise literally from nothing. It's clear then that the classic CTs like the blood libel will also be easily accepted by the people with such a primitive mindset ready to believe anything.

Then there's the fact that the Internet made the resurrection and proliferation of the long-ago forgotten antisemitic literature possible. You can easily find both the original old books with the accusations and their ready translations by the modern neo-Nazis. Something similar is happening with the modern Flat-Earth movement, which was only made possible by the internet.